Branching form of the resolvent at thresholds for multi-dimensional discrete Laplacians
Branching form of the resolvent at thresholds for multi-dimensional discrete Laplacians
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多维离散拉普拉斯算子阈值处求解的分支形式
DOI：
10.1016/j.jfa.2019.05.018
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发表时间：
2019
影响因子：
1.7
通讯作者：
Jensen Arne
中科院分区：
文献类型：
--
作者：
Ito Kenichi;Jensen Arne
我们考虑zd上的离散拉普拉斯算子，并计算其解在连续谱中嵌入的阈值周围和端点处的渐近展开式。证明了当d为奇数时解具有平方根分支，当d为偶数时解具有对数分支，并得到了涉及Lauricella超几何函数的这些分支部分的显式表达式。为了分析一般形式的非退化阈值，我们使用了一个基本的分步展开过程，较少依赖于特殊函数。
We consider the discrete Laplacian on Z d, and compute asymptotic expansions of its resolvent around thresholds embedded in continuous spectrum as well as those at end points. We prove that the resolvent has a square-root branching if d is odd, and a logarithm branching if d is even, and, moreover, obtain explicit expressions for these branching parts involving the Lauricella hypergeometric function. In order to analyze a non-degenerate threshold of general form we use an elementary step-by-step expansion procedure, less dependent on special functions.